Revision: Class 11 >> Oscillations NEET (UG) Oscillations

The particular case of forced oscillations in which the frequency of the driving force equals the natural frequency of the oscillator, and the amplitude of oscillations is very large — such oscillations are called resonant oscillations and the phenomenon is called resonance.

The motion which repeats itself after equal intervals of time is called periodic motion.

The periodic (to and fro) and bounded motion of a body about a fixed point is called oscillatory motion.

When a body, capable of oscillation, is slightly displaced from its position of equilibrium and left to itself, it starts oscillating with a frequency of its own — such oscillations are called free oscillations.

The oscillations in which the amplitude decreases gradually with the passage of time are called damped oscillations.

When energy is continuously supplied from outside to an oscillating system at the same rate at which energy is lost, so that the amplitude remains constant, such oscillations are called maintained oscillations.

When a body oscillates under the influence of an external periodic force, not with its own natural frequency but with the frequency of the external periodic force, its oscillations are called forced oscillations.

The shortest straight line distance between an object's initial and final positions, represented as Δ\[\vec x\] = \[\vec x_2\] − \[\vec x_1\]​​, is called displacement.

OR

The shortest distance from the initial to the final position of a body undergoing motion — a vector quantity whose direction is always from initial to final position, which does not depend on path but only on initial and final positions, and may be positive, negative, or zero — is called displacement.

The total length of the actual path covered by a body in travelling from its initial to its final position — a scalar quantity that always depends on the path followed, can never be negative, and has SI unit metre (m) — is called distance.

The constant φ in the equation of SHM x = Acos⁡(ωt + ϕ) is called phase constant or initial phase.

The physical quantity which represents the position and direction of motion at any instant of the particle executing SHM is called phase.

Any function that repeats itself at regular intervals of its argument is called a periodic function.

If the restoring force/torque acting on a body in oscillatory motion is always directly proportional to its displacement from the equilibrium position and directed towards it, then the motion is called simple harmonic motion (SHM).

The number of oscillations produced by the particle per second is called frequency. It is denoted by f. SI unit for frequency is s⁻¹ or hertz (Hz).

Mathematically, frequency is related to the time period by f = `1/"T"`

Time period: The time period is defined as the time taken by a particle to complete one oscillation. It is usually denoted by T. For one complete revolution, the time taken is t = T, therefore,

`ω"T"` = 2π ⇒ T = `(2π)/ω`

The periodic functions that can be represented by a sine or cosine curve are called harmonic functions.

The periodic functions which cannot be represented by a single sine or cosine function are called non-harmonic functions.

The number of oscillations completed in unit time interval is called frequency of oscillation.

The rate of angular displacement per unit time is called angular frequency.

The maximum value of displacement of the particle from its equilibrium position is called amplitude.

The distance of the particle from the mean position at any instant is called displacement.

The smallest time interval after which the oscillatory motion gets repeated is called time period.

A heavy but small sized metallic bob suspended by a light, inextensible and flexible string, which performs oscillatory motion, is called a simple pendulum.

An ideal simple pendulum consists a point mass suspended from a perfectly rigid support by weightless, inextensible and perfectly flexible fibre.

An ideal simple pendulum is a heavy particle suspended by a massless, inextensible, flexible string from a rigid support.

The frequency with which a body oscillates freely is called natural frequency.

A simple pendulum whose period of oscillation is exactly two seconds is called a second’s pendulum.

A simple pendulum whose period is two seconds is called a second's pendulum.

The force constant of a spring is a measure of its stiffness — it quantifies the force required to produce a unit displacement in the spring, and is called force constant or spring constant.

If the position at time t₁ is x₁ and at time t₂ is x₂, then
Displacement \[\vec s\] = \[\vec x_2\] - \[\vec x_1\]

In vector form:

\[\Delta\bar{r}=(x_2-x_1)\hat{i}+(y_2-y_1)\hat{j}+(z_2-z_1)\hat{k}\]

\[KE=\frac{1}{2}mv^2\]

\[PE=mgh\]

where,

m is mass

g is gravity

h is height

T = 2π\[\sqrt {\frac {l}{g}}\]

n = \[\frac {1}{2π}\]\[\sqrt {\frac {g}{l}}\]

T = 2π\[\sqrt{\frac {L_s}{g}}\] = 2 seconds

\[g=\frac{4\pi^2L}{T^2}\]

\[E=PE+KE\]

• A simple pendulum is a mass on a string swinging under gravity
• Used to determine acceleration due to gravity (g)
• Energy dissipation can be analysed by plotting the square of amplitude vs. time
• At any point: total mechanical energy = PE + KE
• PE depends on height; KE depends on velocity